Then . Let us show that is submultiplicative, i.e. . Indeed, group variables in packages of .

Assume that for all . So for every , there exist points is close to . Go to an ultrapower to find vectors such that . The equality case in the chain of inequalities leading to shows that all norms and for all choices of signs, . Fix signs . If is a unit functional, , thus . In other words, these functionals give an isometry of span onto .

Otherwise, there exists such that . Then there exists , , such that . Let . Given a family , split it according to norms,

Then

Summing over gives the wanted estimate.

1.4. Ribe’s Theorem

It is our first nonlinear result.

Theorem 5 (Ribe 1976) If is a uniform homeomorphism, then is crudely finitely representable in and is crudely finitely representable in .

Proof: Upgrade the uniform homeomorphism to a bi-Lipschitz map. Use differentiability on finite dimensional subspaces to improve bi-Lipschitz maps to isomorphisms.

This implies that and are not uniformly homeomorphic, an earlier result due to Enflo. Enflo’s proof relies on a metric notion of type, on which Naor and Schechtman, Mendel and Naor have elaborated since.

Definition 6 Let be a metric space. Say has Enflo type if there is a constant such that for all ,

where changes the sign of the -th coordinate.

Remark 1 For a Banach space, Enflo type implies usual type .

1.5. Proof of Ribe’s Theorem

Let be a uniform homeomorphism. Then is large-scale Lipschitz: provided . Go to an ultra product to get rid the the large-scale restriction. This means setting . This is bi-Lischitz: . On every finite dimensional subspace , has points of differentiability. The differential is a linear isomorphism of onto a linear subspace of . So is coarsely finitely representable in , and thus in .

2. -convexity

This goes back to Beck and Giesy. They were lookibg for a law of large numbers for Banach space valued random variables.

Proof: Reduce to the case of symmetric distributions by replacing by where is an independent copy of . This produces the factor . Then apply Jensen’s inequality

It implies that

which tends to , this is a law of large numbers.

2.1. Application: approximation of convex hulls

Corollary 8 Let be a Banach space with type . Let . Let . There exist such that

Proof: Apply the law of large numbers to independent copies of the following random variable . By assumption, there exist , such that . So let with probability .

3. Factorization

The following was one of the first results on isomorphism types.

Theorem 9 (Kwapién) If a Banach space has simultaneously type and cotype , then is is isomorphic to a Hilbert space.

Up to now we have uses Bernoulli averages. Why not use Gaussian averages ? Grouping Bernoulli variables (divided by ) in packages of produces variables which are approximately standard Gaussian (central limit theorem). So we can use Gaussian averages instead.